cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A269576 a(n) = Product_{i=1..n} (4^i - 3^i).

Original entry on oeis.org

1, 7, 259, 45325, 35398825, 119187843775, 1692109818073675, 99792176520894983125, 24195710911432718503470625, 23942309231057283642583777144375, 96180015123706384385790918441966041875
Offset: 1

Views

Author

Bob Selcoe, Mar 02 2016

Keywords

Comments

In general, for sequences of the form a(n) = Product_{i=1..n} j^i-k^i, where j>k>=1 and n>=1: given probability p=(k/j)^n that an outcome will occur at the n-th stage of an infinite process, then r = 1 - a(n)/j^((n^2+n)/2) is the probability that the outcome has occurred at or before the n-th iteration. Here j=4 and k=3, so p=(3/4)^n and r = 1-a(n)/A053763(n+1). The limiting ratio of r is ~ 0.9844550.

Links

Crossrefs

Cf. A005061, A053763.
Cf. sequences of the form Product_{i=1..n}(j^i - 1): A005329 (j=2), A027871 (j=3), A027637 (j=4), A027872 (j=5), A027873 (j=6), A027875 (j=7),A027876 (j=8), A027877 (j=9), A027878 (j=10), A027879 (j=11), A027880 (j=12).
Cf. sequences of the form Product_{i=1..n}(j^i - k^1), k>1: A263394 (j=3, k=2), A269661 (j=5, k=4).

Programs

  • Maple
    seq(mul(4^i-3^i,i=1..n),n=0..20); # Robert Israel, Jun 01 2023
  • Mathematica
    Table[Product[4^i - 3^i, {i, n}], {n, 11}] (* Michael De Vlieger, Mar 07 2016 *)
FoldList[Times,Table[4^n-3^n,{n,20}]] (* Harvey P. Dale, Jul 30 2018 *)
  • PARI
    a(n) = prod(k=1, n, 4^k-3^k); \\ Michel Marcus, Mar 05 2016

Formula

a(n) = Product_{i=1..n} A005061(i).
a(n) ~ c * 2^(n*(n+1)), where c = QPochhammer(3/4) = 0.015545038845451847... . - Vaclav Kotesovec, Oct 10 2016
a(n+3)/a(n+2) - 7 * a(n+2)/a(n+1) + 12 * a(n+1)/a(n) = 0. - Robert Israel, Jun 01 2023

A216206 a(n) = Product_{i=1..n} ((-2)^i-1).

Original entry on oeis.org

1, -3, -9, 81, 1215, -40095, -2525985, 325852065, 83092276575, -42626337882975, -43606743654283425, 89350217747626737825, 365889141676531491393375, -2997729737755822508985921375, -49111806293653640164716349886625, 1609344780436736134557590069434814625
Offset: 0

Views

Author

R. J. Mathar, Mar 12 2013

Keywords

Comments

Signed partial products of A062510. This implies that all terms from a(1) on are multiples of 3.

Crossrefs

Cf. A005329, A015013, A015109, A027871, A062510, A330862.

Programs

  • Maple
    A216206 := proc(n)
        mul( (-2)^i-1, i=1..n) ;
end proc:
  • Mathematica
    Table[(-1)^n QPochhammer[-2, -2, n], {n, 0, 15}] (* Bruno Berselli, Mar 13 2013 *)
Table[Product[(-2)^k-1,{k,n}],{n,0,20}] (* Harvey P. Dale, Oct 21 2024 *)

Formula

A015109(n,k) = a(n)/(a(k)*a(n-k)).
a(n) = (-3)^n*A015013(n) for n>0, a(0)=1. - Bruno Berselli and Alonso del Arte, Mar 13 2013
a(n) ~ (-1)^(floor(n/2)+1) * c * 2^(n*(n+1)/2), where c = Product_{k>=1} (1 - 1/(-2)^k) = 1.21072413030105918013... (A330862). - Amiram Eldar, Aug 10 2025

A269661 a(n) = Product_{i=1..n} (5^i - 4^i).

Original entry on oeis.org

1, 9, 549, 202581, 425622681, 4907003889249, 302963327126122509, 98490045052104040328301, 166544794872251942218390753281, 1451779137596368920662880897497387769, 64798450159010700654830227323217753649135349
Offset: 1

Views

Author

Bob Selcoe, Mar 02 2016

Keywords

Crossrefs

Cf. A005060, A109345.
Cf. sequences of the form Product_{i=1..n}(j^i - 1): A005329 (j=2), A027871 (j=3), A027637 (j=4), A027872 (j=5), A027873 (j=6), A027875 (j=7), A027876 (j=8), A027877 (j=9), A027878 (j=10), A027879 (j=11), A027880 (j=12).
Cf. sequences of the form Product_{i=1..n}(j^i - k^1), k>1: A263394 (j=3, k=2), A269576 (j=4, k=3).

Programs

  • Magma
    [&*[ 5^k-4^k: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Mar 03 2016
  • Mathematica
    Table[Product[5^i - 4^i, {i, n}], {n, 15}] (* Vincenzo Librandi, Mar 03 2016 *)
Table[5^(Binomial[n + 1, 2]) *QPochhammer[4/5, 4/5, n], {n, 1, 20}] (* G. C. Greubel, Mar 05 2016 *)
FoldList[Times,Table[5^n-4^n,{n,15}]] (* Harvey P. Dale, Aug 28 2018 *)
  • PARI
    a(n) = prod(k=1, n, 5^k-4^k); \\ Michel Marcus, Mar 05 2016

Formula

a(n) = Product_{i=1..n} A005060(i).
a(n) = 5^(binomial(n+1,2))*(4/5;4/5){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Mar 05 2016
a(n) ~ c * 5^(n*(n+1)/2), where c = QPochhammer(4/5) = 0.00336800585242312126... . - Vaclav Kotesovec, Oct 10 2016

A320354 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = Product_{j=1..n} (k^j - 1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 16, 21, 0, 1, 4, 45, 416, 315, 0, 1, 5, 96, 2835, 33280, 9765, 0, 1, 6, 175, 11904, 722925, 8053760, 615195, 0, 1, 7, 288, 37625, 7428096, 739552275, 5863137280, 78129765, 0, 1, 8, 441, 98496, 48724375, 23205371904, 3028466566125, 12816818094080, 19923090075, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 11 2018

Keywords

Examples

			Square array begins:
  1,     1,        1,          1,            1,             1,  ...
  0,     1,        2,          3,            4,             5,  ...
  0,     3,       16,         45,           96,           175,  ...
  0,    21,      416,       2835,        11904,         37625,  ...
  0,   315,    33280,     722925,      7428096,      48724375,  ...
  0,  9765,  8053760,  739552275,  23205371904,  378832015625,  ...

Crossrefs

Columns k=1..12 give A000007, A005329, A027871, A027637, A027872, A027873, A027875, A027876, A027877, A027878, A027879, A027880.
Cf. A069777, A225816.

Programs

  • Mathematica
    Table[Function[k, Product[k^j - 1, {j, 1, n}]][m - n + 1], {m, 0, 9}, {n, 0, m}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[k^(i (i + 1)/2) x^i/Product[(1 + k^j x), {j, 0, i}], {i, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 9}, {n, 0, m}] // Flatten

Formula

G.f. of column k: Sum_{i>=0} k^(i*(i+1)/2)*x^i / Product_{j=0..i} (1 + k^j*x).
For asymptotics of column k see comment from Vaclav Kotesovec in A027880.
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